In this paper, we investigate the constrained shortest path problem where the arc resources of the problem are dependent normally distributed random variables. A model is presented to maximize the probability of all constraints, while not exceeding a certain amount. We assume that the rows of the constraint matrix are dependent, so we use a marginal distribution of the Copula functions, instead of the distribution functions and the dependency is driven by an appropriate Archimedean Copula. Then, we transform the joint chance-constrained problems into deterministic problems of second-order cone programming. This is a new approach where considers the dependency between resource consumptions and connects Copulas to stochastic resource constrained shortest path problem (SRCSPP). The results indicate that the effect of marginal probability levels is considerable. Moreover, the linear relaxation of SRCSPP is generally not convex; thus we can use lower and upper bounds of the second-order cone programming approximation to solve the relaxation problem. The experimental results show that the SRCSPP with Copula theory can achieve efficient performance.

中文翻译：

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Copula理论的联合机会约束最短路径问题

在本文中，我们研究了约束的最短路径问题，其中问题的弧资源是依赖于正态分布的随机变量。提出了一个模型来最大化所有约束的概率，同时不超过一定数量。我们假设约束矩阵的行是相关的，因此我们使用Copula函数的边际分布，而不是分布函数，并且依赖关系由适当的Archimedean Copula驱动。然后，将联合机会约束问题转化为二阶锥规划的确定性问题。这是一种新方法，其中考虑了资源消耗之间的依赖关系，并将Copulas与随机资源约束的最短路径问题（SRCSPP）连接起来。结果表明，边际概率水平的影响是可观的。此外，SRCSPP的线性弛豫通常不是凸的；因此，我们可以使用二阶锥规划逼近的上下限来解决松弛问题。实验结果表明，采用Copula理论的SRCSPP可以达到较高的性能。